Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log₆ (36/(√(x+1))

Verified step by step guidance

1. Identify the logarithmic expression to expand: \(\log_6 \left( \frac{36}{\sqrt{x+1}} \right)\).
2. Use the logarithm property for division: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). So rewrite the expression as \(\log_6 36 - \log_6 \sqrt{x+1}\).
3. Recognize that \(\sqrt{x+1}\) can be written as \((x+1)^{1/2}\), so apply the power rule for logarithms: \(\log_b (M^p) = p \log_b M\). This gives \(\log_6 36 - \frac{1}{2} \log_6 (x+1)\).
4. Evaluate \(\log_6 36\) by expressing 36 as a power of 6 if possible. Since \$36 = 6^2\(, rewrite \)\log_6 36\( as \)\log_6 (6^2)$.
5. Use the power rule again to simplify \(\log_6 (6^2)\) to \$2 \log_6 6\(, and since \)\log_6 6 = 1\(, this simplifies to 2. So the expanded form is \)2 - \frac{1}{2} \log_6 (x+1)$.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N) and log_b(M^p) = p * log_b(M). These properties are essential for simplifying and expanding logarithmic expressions.

Simplifying Radicals and Exponents

Understanding how to express radicals as fractional exponents is crucial. For instance, the square root of (x+1) can be written as (x+1)^(1/2). This allows the use of the power rule of logarithms to bring the exponent in front of the logarithm, facilitating expansion and simplification.

Evaluating Logarithms Without a Calculator

Some logarithmic values can be simplified or evaluated exactly by recognizing perfect powers or using known logarithm values. For example, log_6(36) can be simplified since 36 = 6^2, so log_6(36) = 2. This helps in reducing expressions to simpler forms without a calculator.